Note that. No office hours on Monday October 15 or Wednesday October 17. No class Tuesday October 16

Transcription

1 advprobnotes101207b Page 1 CDFS and coping with random variables that are not purely discrete nor absolutely continuouss Friday, October 12, :10 PM Note that No office hours on Monday October 15 or Wednesday October 17 No class Tuesday October 16 Homework 2 is due Thursday, October 18 at 5 PM. Turn them into my mailbox in Amos Eaton 301 or to my office Amos Eaton 310. Recall from last time that we can work with computing statistics of arbitrary random variables (which need not be discrete nor absolutely continuous) through the Lebesgue integration framework, which is abstract but essentially completely general.

2 For other reasonable functions g (measurable and integrable) Compute the Lebesgue integral by approximating g by simple functions and passing to the limit. Note that Riemann integration requires laying down a grid in the independent variable space, so it may get confusing or not even possible if the independent variable space is complicated, like a sample space. advprobnotes101207b Page 2

3 advprobnotes101207b Page 3 for more general (measurable and integrable) functions, approximate by simple functions and pass to the limit to compute the Lebesgue integral. And this gives us a way to work with the representation of expectation: Illustrative example of how Lebesgue integration theory makes this possible whereas the standard Riemann integration theory is not so adequate. Consider the probabiliity space consists of the weather around the world over the next month. Consider the random variable Basic reason why the Lebesgue integration framework is used in probability theory is that it can be formulated in terms of discretizing state space, rather than sample space and state space is typically much more concrete and lower dimensional than sample space. Also Lebesgue integration theory unifies discrete and continuous random variables and sample spaces seamlessly. But of course in practical computation the Lebesgue definition is awkward to work with just as definition of Reimann integral is, so we develop special tools for special cases from the general framework. When the state space is one-dimensional, this Lebesgue integration picture can be simplified by recalling the probability distribution of an arbitrary one-dimensonal random variable can be completely characterized by its CDF.

4 advprobnotes101207b Page 4 probability distribution of an arbitrary one dimensonal random variable can be completely characterized by its CDF. For arbitrary simple functions g, take limits of these kinds of simple functions. The fundamental point is that knowing the CDF allows one rigorously to compute any statistic involving that random variable. This is also true in more than one finite dimension but the CDF becomes awkward to work with in

5 advprobnotes101207b Page 5 higher dimensions. The CDF is a useful practical tool for treating hybrid random variables that have both discrete and continuous components. Note on Homework Problem 2.4: By spherical region I mean the interior of ball of radius R (not just the surface). From 09/25, we had a random variable L with the CDF: The CDF is a universally useful object for representing probability distribution for a one-dimensional random variable but we saw that the PDF framework broke down for this example. The problem is that our probability distribution is neither discrete nor absolutely continuous so the specialized (not general) frameworks of PDFs and discrete probability are not adequate. Pop up to the abstract Lebesgue integration level:

6 advprobnotes101207b Page 6 As we take the simple functions to be closer and closer approximations to over the regions where the CDF is differentiable, the contributions from the simple functions:

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8 advprobnotes101207b Page 8 We have argued this informally; on Homework 3 you will have the opportunity to pursue this more rigorously. We can see from these arguments (which can be rigorized) that we can develop another special framework to handle a useful class of hybrid random variables. Consider random variables for which the CDF is piecewise differentiable except at a countable number of points where the CDF has a jump discontinuity with height